Prime Factorization - Grade 6

This page provides comprehensive lessons, examples, and practice questions on Prime Factorization for Grade 6 students. Each practice question includes a step-by-step solution with detailed explanations.

Before diving in, it is highly recommended to review the basics of factors and multiples. Expand the solution tabs below each question to verify your work and understand the underlying logic.

Concepts and Examples

1. Prime Numbers

Definition: A prime number is a whole number strictly greater than 1 that has exactly two distinct factors: 1 and itself.

2. Composite Numbers

Definition: A composite number is a whole number strictly greater than 1 that has more than two factors (it can be divided evenly by numbers other than 1 and itself).

3. Factorization

Definition: To factor a whole number is to write it as a product of two or more whole numbers. Division and multiplication are closely related; the division \(6 \div 3 = 2\) can be rewritten as the multiplication \(6 = 2 \times 3\).

4. Prime Factorization

Definition: Prime factorization is the process of writing a composite whole number strictly as the product of prime numbers.

How to find the prime factorization of 12?

  1. Test the smallest prime number, 2. \(12 \div 2 = 6\). So, \(12 = 2 \times 6\).
  2. Since 6 is composite, test it again. \(6 \div 2 = 3\). So, \(6 = 2 \times 3\).
  3. Combine them: \(12 = 2 \times 2 \times 3\). Since 2 and 3 are prime, the factorization is complete.

How to find the prime factorization of 21?

  1. Test 2: \(21 \div 2\) leaves a remainder. 2 is not a factor.
  2. Test the next prime, 3: \(21 \div 3 = 7\). So, \(21 = 3 \times 7\).
  3. Since 3 and 7 are both prime, the prime factorization is \(3 \times 7\).

Practice Questions

Question 1: Prime vs. Composite

For each number below, decide whether it is a prime or a composite number. Explain your reasoning.

  1. 9
  2. 11
  3. 16
  4. 22
  5. 39
  6. 41
  7. 49
  8. 57
View Solution
Recall: A prime number has exactly 2 factors: 1 and itself. A composite number has 3 or more factors.
  • a) 9: Composite. \(9 = 3 \times 3\). It has 3 factors: 1, 3, 9.
  • b) 11: Prime. It has only 2 factors: 1 and 11.
  • c) 16: Composite. \(16 = 4 \times 4 = 2 \times 8\). It has 5 factors: 1, 2, 4, 8, 16.
  • d) 22: Composite. \(22 = 2 \times 11\). It has 4 factors: 1, 2, 11, 22.
  • e) 39: Composite. \(39 = 3 \times 13\). It has 4 factors: 1, 3, 13, 39.
  • f) 41: Prime. It has only 2 factors: 1 and 41.
  • g) 49: Composite. \(49 = 7 \times 7\). It has 3 factors: 1, 7, 49.
  • h) 57: Composite. \(57 = 3 \times 19\). It has 4 factors: 1, 3, 19, 57.

Question 2: Identifying Prime Factorizations

Which of the following mathematical statements represents a correct prime factorization? Explain why or why not.

  1. \(8 = 2 \times 4\)
  2. \(10 = 2 \times 5\)
  3. \(20 = 2 \times 10\)
  4. \(30 = 2 \times 3 \times 5\)
  5. \(38 = 2 \times 19\)
  6. \(42 = 2 \times 3 \times 7\)
  7. \(56 = 2 \times 2 \times 14\)
  8. \(75 = 3 \times 25\)
  9. \(80 = 2 \times 2 \times 2 \times 2 \times 5\)
  10. \(100 = 2 \times 2 \times 25\)
View Solution
Recall: A true prime factorization must consist of only prime numbers multiplied together.
Equation Is it a Prime Factorization? Reason
a) \(8 = 2 \times 4\) No The factor 4 is a composite number.
b) \(10 = 2 \times 5\) Yes Both 2 and 5 are prime numbers.
c) \(20 = 2 \times 10\) No The factor 10 is a composite number.
d) \(30 = 2 \times 3 \times 5\) Yes All factors (2, 3, 5) are prime.
e) \(38 = 2 \times 19\) Yes Both 2 and 19 are prime.
f) \(42 = 2 \times 3 \times 7\) Yes All factors (2, 3, 7) are prime.
g) \(56 = 2 \times 2 \times 14\) No The factor 14 is a composite number.
h) \(75 = 3 \times 25\) No The factor 25 is a composite number.
i) \(80 = 2 \times 2 \times 2 \times 2 \times 5\) Yes All factors (2 and 5) are prime.
j) \(100 = 2 \times 2 \times 25\) No The factor 25 is a composite number.

Question 3: Calculate the Prime Factorization

Determine the prime factorization for each of the following whole numbers. You may use exponents to simplify your final answer.

  1. 8
  2. 18
  3. 24
  4. 45
  5. 63
  6. 88
  7. 96
View Solution

Prime factorization is carried out by successive divisions using prime numbers (2, 3, 5, 7...):

  • a) 8
    \(8 = 2 \times 4 \rightarrow \mathbf{2 \times 2 \times 2}\) (or \(2^3\))
  • b) 18
    \(18 = 2 \times 9 \rightarrow \mathbf{2 \times 3 \times 3}\) (or \(2 \times 3^2\))
  • c) 24
    \(24 = 2 \times 12 = 2 \times 2 \times 6 \rightarrow \mathbf{2 \times 2 \times 2 \times 3}\) (or \(2^3 \times 3\))
  • d) 45
    \(45 = 3 \times 15 \rightarrow \mathbf{3 \times 3 \times 5}\) (or \(3^2 \times 5\))
  • e) 63
    \(63 = 7 \times 9 \rightarrow \mathbf{3 \times 3 \times 7}\) (or \(3^2 \times 7\))
  • f) 88
    \(88 = 2 \times 44 = 2 \times 2 \times 22 \rightarrow \mathbf{2 \times 2 \times 2 \times 11}\) (or \(2^3 \times 11\))
  • g) 96
    \(96 = 2 \times 48 = 2 \times 2 \times 24 = 2 \times 2 \times 2 \times 12 = 2 \times 2 \times 2 \times 2 \times 6 \rightarrow \mathbf{2 \times 2 \times 2 \times 2 \times 2 \times 3}\) (or \(2^5 \times 3\))

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